Deformations via Simplicial Deformation Complexes

TL;DR

This paper introduces Simplicial Deformation Complexes (SDCs) as a new framework for deformation problems, applicable across all characteristics and capable of handling various algebraic structures.

Contribution

It develops the theory of SDCs as an alternative to DGLAs, providing canonical constructions for multiple deformation problems in characteristic zero and beyond.

Findings

SDCs are equivalent to DGLAs in characteristic zero

SDCs can be constructed canonically for various deformation problems

SDCs work in all characteristics

Abstract

There has long been a philosophy that every deformation problem in characteristic zero should be governed by a differential graded Lie algebra (DGLA). In this paper, the theory of Simplicial Deformation Complexes (SDCs) is developed, as an alternative to DGLAs. These work in all characteristics, and for many problems can be constructed canonically. In particular, SDCs are constructed for the problems of deforming an arbitrary scheme, of deforming a Hopf algebra, and of deforming a representation of the fundamental group. In characteristic zero, SDCs and DGLAs are equivalent.